A Decomposition Theorem for the Translation-Invariant Subspace of a Canonical Differential Operator

1972 ◽  
pp. 120-123
Author(s):  
M. V. Buslaeva
Keyword(s):  
Differential Operator ◽  
Invariant Subspace ◽  
Decomposition Theorem ◽  
Translation Invariant ◽  
Translation Invariant Subspace

Two notes on spectral synthesis for discrete Abelian groups

1965 ◽  
Vol 61 (3) ◽  
pp. 617-620 ◽  
Author(s):  
R. J. Elliott
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Fourier Transforms ◽  
Spectral Synthesis ◽  
Division Problem ◽  
Exponential Functions ◽  
Translation Invariant ◽  
Annihilator Ideal ◽  
The Fourier Transform ◽  
Translation Invariant Subspace

Introduction. Spectral synthesis is the study of whether functions in a certain set, usually a translation invariant subspace (a variety), can be synthesized from certain simple functions, exponential monomials, which are contained in the set. This problem is transformed by considering the annihilator ideal in the dual space, and after taking the Fourier transform the problem becomes one of deciding whether a function is in a certain ideal, that is, we have a ‘division problem’. Because of this we must take into consideration the possibility of the Fourier transforms of functions having zeros of order greater than or equal to 1. This is why, in the original situation, we study whether varieties are generated by their exponential monomials, rather than just their exponential functions. This viewpoint of the problem as a division question, of course, perhaps throws light on why Wiener's Tauberian theorem works, and is implicit in the construction of Schwartz's and Malliavin's counter examples to spectral synthesis in L1(G) (cf. Rudin ((4))).

scholarly journals Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators

2020 ◽  
Vol 14 (8) ◽  
Author(s):  
Ryan O’Loughlin
Keyword(s):  
Hardy Space ◽  
Toeplitz Operator ◽  
Invariant Subspace ◽  
Toeplitz Operators ◽  
Invariant Subspaces ◽  
Decomposition Theorem ◽  
Truncated Toeplitz Operator ◽  
Finite Defect ◽  
Vector Valued

AbstractIn this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invariant subspaces with a finite defect. This is a powerful result which allows us to gain insight in to the structure of scalar subspaces of the Hardy space using vector-valued Hardy space techniques. These results have far reaching applications, in particular they allow us to develop an all encompassing approach to the study of the kernels of: the Toeplitz operator, the truncated Toeplitz operator, the truncated Toeplitz operator on the multiband space and the dual truncated Toeplitz operator.

scholarly journals On Some Normality-Like Properties and Bishop's Property () for a Class of Operators on Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Sid Ahmed Ould Ahmed Mahmoud
Keyword(s):  
Invariant Subspace ◽  
Hilbert Spaces ◽  
Invariant Property ◽  
Translation Invariant ◽  
Scalar Operator

We prove some further properties of the operator (-power quasinormal, defined in Sid Ahmed, 2011). In particular we show that the operator satisfying the translation invariant property is normal and that the operator is not supercyclic provided that it is not invertible. Also, we study some cases in which an operator is subscalar of order ; that is, it is similar to the restriction of a scalar operator of order to an invariant subspace.

Tensor Algebras, Induced Representations, and the Wold Decomposition

1999 ◽  
Vol 51 (4) ◽  
pp. 850-880 ◽  
Author(s):  
Paul S. Muhly ◽  
Baruch Solel
Keyword(s):  
Hilbert Space ◽  
Invariant Subspace ◽  
Decomposition Theorem ◽  
Wold Decomposition ◽  
Induced Representations ◽  
Tensor Algebras ◽  
Beurling’S Theorem

AbstractOur objective in this sequel to [18] is to develop extensions, to representations of tensor algebras over C*-correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling’s theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras.

scholarly journals Bounded solutions of nonlinear Cauchy problems

2002 ◽  
Vol 7 (12) ◽  
pp. 637-661 ◽  
Author(s):  
Josef Kreulich
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Existence Of Solutions ◽  
Continuous Functions ◽  
Almost Periodicity ◽  
Cauchy Problems ◽  
Bounded Solutions ◽  
Translation Invariant ◽  
Uniformly Continuous ◽  
Translation Invariant Subspace

For a given closed and translation invariant subspaceYof the bounded and uniformly continuous functions, we will give criteria for the existence of solutionsu∈Yto the equationu′(t)+A(u(t))+ωu(t)∍f(t),t∈ℝ, or of solutionsuasymptotically close toYfor the inhomogeneous differential equationu′(t)+A(u(t))+ωu(t)∍f(t),t>0,u(0)=u0, in general Banach spaces, whereAdenotes a possibly nonlinear accretive generator of a semigroup. Particular examples for the spaceYare spaces of functions with various almost periodicity properties and more general types of asymptotic behavior.

scholarly journals On the mild solutions of higher-order differential equations in Banach spaces

2003 ◽  
Vol 2003 (15) ◽  
pp. 865-880 ◽  
Author(s):  
Nguyen Thanh Lan
Keyword(s):  
Differential Equation ◽  
Banach Spaces ◽  
Mild Solutions ◽  
Higher Order ◽  
Almost Periodicity ◽  
Abstract Differential Equation ◽  
Translation Invariant ◽  
Definition Of ◽  
Translation Invariant Subspace ◽  
Higher Order Differential Equations

For the higher-order abstract differential equationu(n)(t)=Au(t)+f(t),t∈ℝ, we give a new definition of mild solutions. We then characterize the regular admissibility of a translation-invariant subspaceℳofBUC(ℝ,E)with respect to the above-mentioned equation in terms of solvability of the operator equationAX−X𝒟n=C. As applications, periodicity and almost periodicity of mild solutions are also proved.

Some results in spectral synthesis

1965 ◽  
Vol 61 (2) ◽  
pp. 395-424 ◽  
Author(s):  
Robert J. Elliott
Keyword(s):  
Spectral Synthesis ◽  
Related Result ◽  
Convolution Equation ◽  
Continuous Functions ◽  
Exponential Polynomial ◽  
Space Of Continuous Functions ◽  
Translation Invariant ◽  
Famous Paper ◽  
Translation Invariant Subspace ◽  
Homogeneous Convolution Equation

For the group of real numbers R, an exponential monomial is defined as a function of the form xr(−ixz), for some non-negative integer r and some complex number z. Similarly, an exponential polynomial is a function P(x) exp (−ixz), for a polynomial P. In a now famous paper ((15)), Schwartz proved that every closed translation invariant subspace (variety) of the space of continuous functions on R is determined by the exponential monomials it contains. His techniques do not generalize to groups other than R as they use the theory of functions of one complex variable. A shorter proof of this result, using the Carleman transform of a function, was given by Kahane in his thesis ((9)). Ehrenpreis ((5)) proved results similar to those of Schwartz for certain varieties in the space of analytic functions of n complex variables, and Malgrange ((13)) proved the related result that any solution in ℰ(Rn) (for the notation see (16)) of the homogeneous convolution equation μ*f = 0, for some μ∈ℰ′, belongs to the closure of the exponential polynomial solutions of the equation.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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